# Closed form for $\sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+…}}}$

Is there a close form for the following nested radical?

$$\sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...}}}$$

It converges and
$$\quad \quad \lim_{n \to\infty} \sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...+\sqrt[n!]{n^n}}}}=1.8430759846682...$$

Is this number algebraic or transcendental?

• If it has a closed form, then $e$ is likely involved. – Vincenzo Oliva Jul 2 '15 at 21:21
• I honestly doubt that a closed form will be found, but of course you never know. – Wojowu Jul 2 '15 at 21:30
• Is there any context from which the above constant arises? If this is just a radical made on a spot, I see no reason for it to have a closed form. – Wojowu Jul 2 '15 at 21:39
• I conjecture that almost surely 1) it does not have a closed form 2) it is trancendental 3) nobody will be able to provide an answer (proof) in one way or another to either of the two questions. – Winther Jul 2 '15 at 21:49

• @QuantumDot: Never mind, I forgot to extract the square root. My sequence started $\sqrt[1!]{1^1+\sqrt[2!]{2^2+\ldots}}$ – Lucian Nov 27 '15 at 10:44